We have described in the last section, how the early universe looks like, and thereby what kind of initial conditions are suitable for cosmological simulations. In this section we want to review shortly the evolution equations of the dark matter fluid. Therefore we will first derive the equations of motion for a single particle that is surrounded by some mass distribution ρ(x). Afterwards we will discuss the collisionless Boltzmann equation, which is the most fundamental description of the collisionless dark matter fluid. Combined with the Poisson equation this forms the Vlasov-Poisson system. The underlying assumption is that dark matter can be treated as a collisionless fluid.
It is quite easy to see that dark matter can be treated as a fluid. By this we mean that it is well described by some continuous phase space densityf(x,v). It consists of a gigantic number of particles when compared to the cosmological scales that we will speak of in this thesis. For example if it is made out of am ∼100 GeV particle (e.g. a WIMP), the number density is approximatelyρ0/m∼0.03 m−3. That corresponds to∼1048particles in a cubic parsec volume. Therefore at cosmological length scales (kilo-parsecs to Giga-parsecs) dark matter can be considered as a continuous fluid.
The treatment of dark matter as collisionless is less obvious. It depends on the nature of dark matter whether it has self-interactions and how strong these self-interactions are.
However, for most considered dark matter models self-interactions are very weak. Further, the self-interactions have been constrained by astrophysical observations like the Bullet cluster to be quite small (Robertson et al., 2017). We want to point out here that measuring the degree of self-interaction from astrophysical observations might be a way to learn more about its nature. However, so far it is consistent with a completely collisionless picture.
For simplicity we will assume dark matter as collision-less for the remainder of this thesis.
Therefore dark matter can be treated as a self-gravitating collisionless fluid.
2.2.1 The equations of motion
The correct way of deriving the equations of motion in an expanding universe is to derive a general relativistic description of its metric and then set up the geodesic equation. However, this is not very intuitive and makes it quite hard to see how Hamiltonian phase space dynamics generalize into the comoving frame. Assuming that all involved velocities are non-relativistic, locally the equations of motion have to revert to a Newtonian description.
To get an intuitive picture of what changes when viewing the world from the comoving
2.2 Evolution equations 31 frame, we start from the Newtonian equations of motion and transform them to comoving coordinates. For this, we take the Friedman equations as given and assume a universe which contains only matter (Λ = 0). We will discuss afterwards what has to change if a cosmological constant is included.
In the Newtonian frame the equations of motion of a single particle are given by
¨r=−∇rφr (2.32)
∇2rφ= 4πGρr (2.33)
wherer are its physical coordinates, φr is the Newtonian potential, ρr is the mass density,
∇r denotes a gradient operator with respect to physical coordinates and dotted variables denote time derivatives.
We can choose to transform the coordinates r into any other frame and reexpress the equations of motion in that frame. For an expanding universe a convenient choice is
x:= r
a (2.34)
v:=xa˙ 2 = ˙ra−ra˙ (2.35)
∇x =a∇r (2.36)
where x are comoving coordinates and v is a velocity variable (where peculiar velocities are given by v/a). With the new coordinates we can reexpress the equations of motions as
˙ x= v
a2 (2.37)
˙
v=¨ra−r¨a (2.38)
=:−∇xφc
a (2.39)
∇2xφ=a2∇r(−¨ra+r¨a) (2.40)
=a3
4πGρr+ 3a¨ a
(2.41) where we have defined the peculiar potential φ. The choices forx,vandφ seem somewhat arbitrary up to this point. However, if the function a(t) is chosen as the expansion factor of the universe we can use the second Friedmann equation (1.3) to eliminate ¨a/a from the new Poisson equation:
∇2xφ=a3(4πGρr−4πGρbg,m) (2.42)
= 4πGa3ρm(ρr−ρm)
ρm (2.43)
= 4πGρ0δ (2.44)
where we have defined the relative overdensityδand used that matter dilutes asρm =a−3ρ0 whereρ0 is the mean matter density of the universe ata= 1. We summarize the equations of motion and the Poisson equation:
˙ x:= v
a2 (2.45)
˙
v:=−∇xφ
a (2.46)
∇2xφ= 4πGρ0δ= 3
2Ω0H02δ (2.47)
(cf. Peebles, 1980) where we also gave an alternative version for the Poisson equation by re-expressing the mean matter density as ρ0 =ρcritΩ0 = 3H02Ω0/(8πG). The equations of motion only appear that simple because of the particular choices of x, v and φ. This coordinate choice ensures that xis constant as a function of time if vis zero,v is constant if the mass distribution is homogeneous δ = 0 and φ is constant if δ is constant. In other coordinate choices (e.g. when using the peculiar velocity u = v/a as velocity variable) additional time and/or coordinate dependencies can arise.
A cosmological constant in the Newtonian Frame
In our derivation we assumed Λ = 0 and neglected radiation (p = 0). However, the final equations of motion are still valid in a universe with a cosmological constant. What changes when including a cosmological constant are the Newtonian equations of motion that we started from. For a non-zero cosmological constant they must read
¨r=−∇rφr+Λc2
3 r (2.48)
if we do not absorb the cosmological constant into the definition of the potential. In the Newtonian frame the cosmological constant acts as an additional distance dependent force.
In the case of a negligible gravitational force ∇φr = 0 the physical distance between two test particles would grow exponentially over time:
r(t) =r0exp
rΛc2 3 t
!
(2.49) However in the case of strong gravitational forces k∇rφrk Λc2r the effect of the cos-mological constant can generally be neglected. For example let us consider the effect of the additional force due to the cosmological constant in the case of a test-particle in an (otherwise) Keplerian potential:
¨
r=−M G r2 + Λc2
3 r (2.50)
This system has an effective attractive central potential (assuming Λ is small). Therefore the test particle does not spiral outwards. Instead it has a slightly modified bound orbit.
Interestingly in the case of the Keplerian potential orbits do not close anymore, but instead
2.2 Evolution equations 33 have a precession. However that precession is far too small to be measured. For example in the case of Mercury it is more than 10 orders of magnitude smaller than the measurement uncertainty (Adkins et al., 2007). Post-Newtonian corrections are much larger than the effect of the cosmological constant in our solar system. However, it is not completely unthinkable that in a distant future we could also measure the cosmological constant by its effect on closed orbits.
2.2.2 The Vlasov-Poisson system
Dark matter is collisionless and as a fluid it can be well described by a continuous phase space density f(x,v). In the comoving frame that we are using here this phase space density is conserved along trajectories. This is described by the collisionless Boltzmann equation
df(x,v, t)
dt = ∂f
∂t + ∂f
∂x· dx dt + ∂f
∂v · dv
dt = 0 (2.51)
The combination with the Poisson equation (2.47) and using the equations of motion from (2.45) - (2.46) leads to the Vlasov Poisson system
0 = ∂f
∂t +∇xf · v
a2 −∇vf ·∇xφ
a (2.52)
ρ(x, t) =mx Z
f(x,v, t)d3v (2.53)
∇2xφ= 4πG(ρ−ρ0) (2.54)
Note that the choice of position and velocity variables ensures that the associated phase space density is conserved dfdt = 0. If e.g. the peculiar velocity was chosen as velocity variable this would not be the case. Interestingly more or less the same system appears in the description of collisionless plasmas, since the electrodynamical potential also follows the Poisson equation.
2.2.3 Impact of baryons
All cosmological gravity-only simulations (without self-interactions) try to solve the evolu-tion of this system (in some approximate manner). If baryonic interacevolu-tions are included, the evolution of the collisionless dark matter part of the system still stays equivalent, only the density receiving an additional time dependent component
ρ(x, t) = Z
fdm(x,v, t)d3v+ρb(x, t) (2.55) where the baryon density ρb(x, t) evolves according to a different set equations, since it is collisional. In most cases it is assumed that the baryons are in local thermodynamic
equilibrium and therefore can be approximated by hydrodynamic equations. These evolve projected distribution functions that are only a function of spacex, but not velocity (like f). However, we will not further worry about the influence of baryons in this thesis, but instead focus on the gravity-only or dark matter only case ρb = 0. We just want to note here that all qualitative statements in this thesis about the phase space distribution of dark matter still hold in the case that baryons were included. That is so, because dark matter still follows the collisionless Boltzmann equation and the Baryons have only a very indirect impact by modifying the source term in the Poisson equation. However, baryons can modify quantitative results.